Question

If $f\left(x\right)=\log \left(\frac{x^2+ab}{x(a+b)}\right)$, then the value of ${}^{\prime }{C}^{\prime }$ for which $f^{\prime }\left(c\right)=0$ in $[a,b]$

• A. $C=\pm \sqrt{\frac{a}{b}}$
• B. $C=\pm \sqrt{ab}$
• C. $C=\pm \sqrt{\frac{b}{a}}$
• D. $noneofthese$

Answer 1

The correct option is B $C=\pm \sqrt{ab}$

$f\left(x\right)=\log \left(\frac{x^2+ab}{x(a+b)}\right)$

$f^{\prime }\left(x\right)=\frac{-x(a+b)}{x^2+ab}\left[\frac{1-\frac{ab}{x^2}}{(a+b)}\right]$

$=\frac{-x}{x^2+ab}\frac{(x^2-ab)}{x^2}$

$\Rightarrow \frac{-(x^2-ab)}{x(x^2+ab)}=0$

$x=\pm \sqrt{ab}$